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Miha Kralj
98 lines
4.4 KiB
Markdown
98 lines
4.4 KiB
Markdown
# MINUS_DI: Minus Directional Indicator
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> *-DI isolates downward directional thrust as a fraction of true range — the bearish arm of Wilder's directional system.*
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| Property | Value |
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| ---------------- | -------------------------------- |
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| **Category** | Dynamic |
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| **Inputs** | OHLCV bar (TBar) |
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| **Parameters** | `period` (default 14) |
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| **Outputs** | Single series |
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| **Output range** | 0 to 100 |
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| **Warmup** | `period` bars |
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| **PineScript** | [minusdi.pine](minusdi.pine) |
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- The Minus Directional Indicator measures the strength of downward price movement relative to true range.
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- **Similar:** [PlusDi](../plusdi/PlusDi.md), [ADX](../adx/Adx.md) | **Complementary:** ADX for trend strength | **Trading note:** Minus Directional Indicator; measures downward movement strength. Part of Wilder's DM system.
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- Validated against TA-Lib, Skender, and Dx equivalence.
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The Minus Directional Indicator (-DI) is one component of J. Welles Wilder Jr.'s Directional Movement System. It quantifies the fraction of recent true range attributable to downward price extension. The computation smooths both -DM (minus directional movement) and TR (true range) with Wilder's RMA ($\alpha = 1/N$), then divides: $-DI = 100 \times \text{Smooth}(-DM) / \text{Smooth}(TR)$. When -DI rises, downward price pressure is increasing. When -DI crosses above +DI, it signals a potential bearish trend. The -DI line is commonly plotted alongside +DI to visualize directional balance.
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## Historical Context
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J. Welles Wilder Jr. introduced the Directional Movement System in *New Concepts in Technical Trading Systems* (1978). The system decomposes price range into directional components. +DI and -DI are the normalized indicators from which DX and ADX are derived. While most traders focus on ADX for trend strength, +DI and -DI remain essential for determining trend *direction* — a bearish signal occurs when -DI crosses above +DI, bullish when +DI crosses above -DI.
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## Architecture & Physics
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### 1. Minus Directional Movement
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$$\text{UpMove} = H_t - H_{t-1}, \quad \text{DownMove} = L_{t-1} - L_t$$
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$$-DM = \begin{cases} \text{DownMove} & \text{if DownMove} > \text{UpMove and DownMove} > 0 \\ 0 & \text{otherwise} \end{cases}$$
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### 2. True Range
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$$TR = \max(H_t - L_t,\; |H_t - C_{t-1}|,\; |L_t - C_{t-1}|)$$
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### 3. Wilder Smoothing (RMA)
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$$-DM_{\text{smooth}} = \text{RMA}(-DM, N), \quad TR_{\text{smooth}} = \text{RMA}(TR, N)$$
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### 4. Minus Directional Indicator
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$$-DI = 100 \times \frac{-DM_{\text{smooth}}}{TR_{\text{smooth}}}$$
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When $TR_{\text{smooth}} = 0$ (no price movement), -DI = 0.
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### 5. Complexity
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- **Time:** $O(1)$ per bar — all RMA updates are recursive
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- **Space:** $O(1)$ — scalar state only (delegates to Dx)
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- **Warmup:** $N$ bars
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## Mathematical Foundation
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### Parameters
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| Symbol | Parameter | Default | Constraint |
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|--------|-----------|---------|------------|
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| $N$ | period | 14 | $N \geq 2$ |
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### Interpretation
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| -DI Value | Signal |
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|-----------|--------|
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| Rising -DI | Strengthening downward movement |
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| -DI > +DI | Bears dominate; potential downtrend |
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| -DI crossover above +DI | Bearish signal |
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| High -DI (>40) | Strong downward momentum |
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-DI measures directional *strength*, not absolute direction. Compare +DI vs -DI for directional bias: if $-DI > +DI$, the trend is down.
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## Performance Profile
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### Operation Count (Streaming Mode)
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-DI is a thin wrapper around Dx. The per-bar cost is identical to Dx (one property extraction after Dx completes its update).
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**Post-warmup steady state (per bar):**
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| Operation | Count | Cost (cycles) | Subtotal |
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| :--- | :---: | :---: | :---: |
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| Dx.Update (full pipeline) | 1 | 75 | 75 |
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| Property extraction | 1 | 1 | 1 |
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| **Total** | **2** | — | **~76 cycles** |
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### Quality Metrics
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| Metric | Score | Notes |
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| :--- | :---: | :--- |
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| **Accuracy** | 9/10 | Exact Dx delegation; FMA-precise RMA smoothing |
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| **Timeliness** | 7/10 | N-bar warmup; responds to bar-level changes |
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| **Smoothness** | 7/10 | Single RMA layer; moderate noise suppression |
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| **Noise Rejection** | 7/10 | Wilder smoothing filters transient spikes |
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## Resources
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- Wilder, J.W. — *New Concepts in Technical Trading Systems* (Trend Research, 1978)
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- PineScript reference: `minusdi.pine` in indicator directory |